Écrit avec Dominique Delande et en collaboration avec l’équipe de Phillips (groupe Laser Cooling and Trapping du nist, Gaithersburg), Winfried Hensinger en particulier, qui était en post-doc directement en chargé de l’expérience, nous analysons en détail les résultats en nous appuyant sur des simulations de l’équation de Gross-Pitaevskii, d’une part, et sur un analyse de la dynamique de Floquet d’autre part.
Analysis of dynamical tunneling experiments with a Bose-Einstein condensate W. K. Hensinger,1,2,* A. Mouchet,3,†P. S. Julienne,2D. Delande,4N. R. Heckenberg,1and H. Rubinsztein-Dunlop1
1
Centre for Biophotonics and Laser Science, Department of Physics, The University of Queensland, Brisbane, Queensland 4072, Australia
2
National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
3
Laboratoire de Mathématique et de Physique Théorique (CNRS UMR 6083), Avenue Monge, Parc de Grandmont, 37200 Tours, France
4
Laboratoire Kastler-Brossel (CNRS UMR 8552), Université Pierre et Marie Curie, 4 place Jussieu, F-75005 Paris, France (Received 26 September 2003; revised manuscript received 5 April 2004; published 15 July 2004)
Dynamical tunneling is a quantum phenomenon where a classically forbidden process occurs that is prohib-ited not by energy but by another constant of motion. The phenomenon of dynamical tunneling has been recently observed in a sodium Bose-Einstein condensate. We present a detailed analysis of these experiments using numerical solutions of the three-dimensional Gross-Pitaevskii equation and the corresponding Floquet theory. We explore the parameter dependency of the tunneling oscillations and we move the quantum system towards the classical limit in the experimentally accessible regime.
DOI: 10.1103/PhysRevA.70.013408 PACS number(s): 42.50.Vk, 32.80.Pj, 05.45.Mt, 03.65.Xp
I. INTRODUCTION
Cold atoms provide a system which is particularly suited to study quantum nonlinear dynamics, quantum chaos, and the quantum-classical borderland. On relevant time scales the effects of decoherence and dissipation are negligible. This allows us to study a Hamiltonian quantum system. Only recently dynamical tunneling was observed in experiments with ultracold atoms [1,2]. “Conventional” quantum tunnel-ing allows a particle to pass through a classical energy bar-rier. In contrast, in dynamical tunneling a constant of motion other than energy classically forbids one to access a different motional state. In our experiments atoms tunneled back and forth between their initial oscillatory motion and the motion 180° out of phase. A related experiment was carried out by Steck, Oskay, and Raizen [3,4] in which atoms tunneled from one unidirectional librational motion into another oppositely directed motion.
Luter and Reichl [5] analyzed both experiments calculat-ing mean momentum expectations values and Floquet states for some of the parameter sets for which experiments were carried out and found good agreement with the observed tunneling frequencies. Averbukh, Osovski, and Moiseyev [6] pointed out that it is possible to effectively control the tun-neling period by varying the effective Planck’s constant by only 10%. They showed one can observe both suppression due to the degeneracy of two Floquet states and enhancement due to the interaction with a third state in such a small inter-val.
Here we present a detailed theoretical and numerical analysis of our experiments. We use numerical solutions of the Gross-Pitaevskii equation and Floquet theory to analyze
the experiments and to investigate the relevant tunneling dy-namics. In particular we show how dynamical tunneling can be understood in a two and three state framework using Flo-quet theory. We show that there is good agreement between experiments and both Gross-Pitaevskii evolution and Floquet theory. We examine the parameter sensitivity of the tunneling period to understand the underlying tunneling mechanisms. We also discuss such concepts as chaos-assisted and resonance-assisted tunneling in relation to our experimental results. Finally predictions are made concerning what can happen when the quantum system is moved towards the clas-sical limit.
In our experiments a sodium Bose-Einstein condensate was adiabatically loaded into a far detuned optical standing wave. For a sufficient large detuning, spontaneous emission can be neglected on the time scales of the experiments
s160 msd. This also allows us to consider the external
de-grees of freedom only. The dynamics perpendicular to the standing wave are not significant, therefore we are led to an effectively one-dimensional system. The one-dimensional system can be described in the corresponding two-dimensional phase space which is spanned by momentum and position coordinates along the standing wave. Single fre-quency modulation of the intensity of the standing wave leads to an effective Hamiltonian for the center-of-mass mo-tion given by H = px 2 2m+ "Veff 4 f1 − 2« sinsvt + fdgsin2skxd, s1d
where the effective Rabi frequency is Veff= V2/d, V = GÎI / Isatis the resonant Rabi frequency, « is the modulation parameter, v is the modulation angular frequency, G is the inverse spontaneous lifetime,dis the detuning of the stand-ing wave, t is the time, pxis the momentum component of the atom along the standing wave, and k is the wave number. Here I is the spatial-mean of the intensity of the unmodulated standing wave (which is half of the peak intensity) so V *Electronic address: hensinge@umich.edu; Present address:
De-partment of Physics, University of Michigan, 2477 Randall Labo-ratory, 500 East University Ave., Ann Arbor, MI 48109-1120, USA.
†
Electronic address: mouchet@celfi.phys.univ-tours.fr
= GÎIpeak/ 2Isatwhere Isat= hcG / l3is the saturation intensity.
l is the wavelength of the standing wave.f determines the
start phase of the amplitude modulation. Using scaled vari-ables [7] the Hamiltonian is given by
H = p2/2 + 2kf1 − 2« sinst + fdgsin2sq/2d, s2d
where H =s4k2/ mv2dH, q = 2kx, and p = s2k / mvdpx. The driving amplitude is given by
k = vrVeff/v2="k2Veff
2v2m = 4U0vr
v2" , s3d
where vr= "k2/ 2m is the recoil frequency,t = tv is the scaled
time variable, and U0is the well depth. The commutator of scaled position and momentum is given by
fp,qg = i k–, s4d
where the scaled Planck’s constant is k– =8vr/ v. Fork = 1.2
and « = 0.20 the classical Poincaré surface of section is shown in Fig. 1. Two symmetric regular regions can be ob-served aboutsq = 0 , p = 1d and sq = 0 , p = −1d. These regions
correspond to oscillatory motion in phase with the amplitude modulation in each well of the standing wave. In the experi-ment [1,2] atoms are loaded in a period-1 region of regular motion by controling their initial position and momentum and by choosing the starting phase of the amplitude modula-tion appropriately. Classically atoms should retain their mo-mentum state when observed stroboscopically (time step is one modulation period). A distinct signature of dynamical
tunneling is a coherent oscillation of the stroboscopically observed mean momentum as shown in Fig. 2 and reported in Ref. [1].
In Sec. II we introduce the theoretical tools to analyze dynamical tunneling by discussing Gross-Pitaevskii simula-tions and the appropriate Floquet theory. We present a thor-ough analysis of the experiments from Ref. [1] in Sec. III. After showing some theoretical results for the experimental parameters we give a small overview of what to expect when some of the system parameters in the experiments are varied in Sec. IV and give some initial analysis. In Sec. V we point to pathways to analyze the quantum-classical transition for our experimental system and give conclusions in Sec. VI.
II. THEORETICAL ANALYSIS OF THE DYNAMIC EVOLUTION OF A BOSE-EINSTEIN CONDENSATE
A. Dynamics using the Gross-Pitaevskii equation
The dynamics of a Bose-Einstein condensate in a time-dependent potential in the mean-field limit are described by the Gross-Pitaevskii equation [8,9]
i"] Csr,td ] t =
F
− " 2m¹ 2+ Vtrapsr,td + Vsr,td + N4p " 2a m uCsr,tdu2G
Csr,td, s5dwhere N is the mean number of atoms in the condensate, and a is the scattering length with a = 2.8 nm for sodium. Vtrapsrd
is the trapping potential which is turned off during the inter-action with the standing wave and Vsr , td is the
time-FIG. 1. Poincaré section for a classical particle in an amplitude-modulated optical standing wave. Momentum and position (one well of the standing wave) of the particle along the standing wave are plotted stroboscopically with the stroboscopic period being equal to the modulation period. The central region consists of small amplitude motion. Chaos (dotted region) separates this region from two period-1 regions of regular motion (represented in the Poincaré section as sets of closed curves) located left and right of the center along momentum p = 0. Further out in momentum are two stable regions of motion known as librations. At the edges are bands of regular motion corresponding to above barrier motion. It is plotted for modulation parameter « = 0.20 and scaled well depthk = 1.20.
FIG. 2. Stroboscopic mean momentum as a function of the in-teraction time with the modulated standing wave measured in modulation periods, n, for modulation parameter « = 0.29, scaled well depthk = 1.66, modulation frequency v / 2p = 250 kHz, and a
phase shift w = 0.213 2p. (a) and (b) correspond to the two different interaction times n + 0.25 and n + 0.75 modulation periods, respec-tively. Results from the dynamic evolution of the Gross-Pitaevskii equation are plotted as a solid line (circles) and the experimental data are plotted as a dashed line (diamonds).
dependent optical potential induced by the optical standing wave. The Gross-Pitaevskii equation is propagated in time using a standard numerical split-operator, fast Fourier trans-form method. The size of the spatial grid of the numerical simulation is chosen to contain the full spatial extent of the initial condensate (therefore all the populated wells of the standing wave), and the grid has periodic boundary condi-tions at each side (a few unpopulated wells are also included on each side).
To obtain the initial wave function a Gaussian test func-tion is evolved by imaginary time evolufunc-tion to converge to the ground state of the stationary Gross-Pitaevskii equation. Then the standing wave is turned on adiabatically with Vsr , td approximately having the form of a linear ramp. After
the adiabatic turn-on, the condensate wave function is found to be localized at the bottom of each well of the standing wave. The standing wave is shifted and the time-dependent potential now has the form
Vsr,td ="Veff
4 f1 − 2« sinsvt + fdgsin2skx + wd, s6d
where w is the phase shift which is applied to selectively load one region of regular motion [1]. The position representation of the atomic wave functionuCsrdu2just before the modula-tion starts (and after the phase shift) is shown in Fig. 3st
= 0Td.
The Gross-Pitaevskii equation is used to model the ex-perimental details of a Bose-Einstein condensate in an opti-cal one-dimensional lattice. The experiment effectively con-sists of many coherent single-atom experiments. The coherence is reflected in the occurrence of diffraction peaks in the atomic momentum distribution (see Fig. 4). Utilizing
the Gross-Pitaevskii equation, the interaction between these single-atom experiments is modeled by a classical field. Al-though ignoring quantum phase fluctuations in the conden-sate, the wave nature of atoms is still contained in the Gross-Pitaevskii equation and dynamical tunneling is a quantum effect that results from the wave nature of the atoms. The assumption for a common phase for the whole condensate is well justified for the experimental conditions as the time scales of the experiment and the lattice well depth are suffi-ciently small. It will be shown in the following (see Fig. 8) that the kinetic energy is typically of the order of 105Hz which is much larger than the nonlinear term in the Gross-Pitaevskii equation (5) which is on the order of 400 Hz. The experimental results, in particular dynamical tunneling, could therefore be modeled by a single particle Schrödinger equation in a one-dimensional single well with periodic boundary conditions. Nevertheless the Gross-Pitaevskii equation is used to model all the experimental details of a Bose-Einstein condensate in an optical lattice to guarantee maximum accuracy. We will discuss and compare the Gross-Pitaevskii and the Floquet approaches below (last paragraph of Sec. II).
Theoretical analysis of the dynamical tunneling experi-ments will be presented in this paper utilizing numerical so-lutions of the Gross-Pitaevskii equation. Furthermore we will analyze the system parameter space which is spanned by the scaled well depth k, the modulation parameter «, and the
scaled Planck’s constant k–. In fact variation of the scaled Planck’s constant in the simulations allows one to move the quantum system towards the classical limit.
B. Floquet analysis
The quantum dynamics of a periodically driven Hamil-tonian system can be described in terms of the eigenstates of
FIG. 3. Position representation of the atomic wave function as a function of the number of modu-lation periods calculated using the Gross-Pitaevskii equation. The wave packet is plotted strobo-scopically at a phase so it is lo-cated classically approximately at its highest point in the potential well. The position of the standing wave wells is also shown (dotted line). Dynamical tunneling can be observed. At t = 5T most atoms have tunneled into the other period-1 region of regular motion. The position axis is given in Fermi units (scaled by the mean Thomas-Fermi diameter).
the Floquet operator F, which evolves the system in time by one modulation period. In the semiclassical regime, the Flo-quet eigenstates can be associated with regions of regular and irregular motion of the classical map. However, when " is not sufficiently small compared to a typical classical action the phase-space representation of the Floquet eigenstates do not necessarily match with some classical (regular or irregu-lar) structures [10,11]. However, initial states localized at the stable region around a fixed point in the Poincaré section can be associated with superpositions of a small number of Flo-quet eigenstates. Using this state basis, one can reveal the analogy of the dynamical tunneling experiments and conven-tional tunneling in a double well system. Two states of op-posite parity which can be responsible for the observed dy-namical tunneling phenomenon are identified. Floquet states are stroboscopic eigenstates of the system. Their phase space representation therefore provides a quantum analog to the classical stroboscopic phase space representation, the Poincaré map.
Only very few states are needed to describe the evolution of a wave packet that is initially strongly localized on a re-gion of regular motion. A strongly localized wave packet is used in the experiments (strongly localized in each well of the standing wave) making the Floquet basis very useful. In contrast, describing the dynamics in momentum or position representation requires a large number of states so that some of the intuitive understanding which one can obtain in the Floquet basis is impossible to gain. For example, the tunnel-ing period can be derived from the quasi-eigenenergies of the relevant Floquet states, as will be shown below.
For an appropriate choice of parameters the phase space exhibits two period-1 fixed points, which for a suitable Poincaré section lie on the momentum axis at ±p0, as in Ref.
[1]. For certain values of the scaled well depthk and
modu-lation parameter « there are two dominant Floquet statesuf±l
that are localized on both fixed points but are distinguished by being even or odd eigenstates of the parity operator that changes the sign of momentum. A state localized on just one fixed point is therefore likely to have dominant support on an even or odd superposition of these two Floquet states:
ucs±p0dl < suf+l ± uf−ld/Î2. s7d
The stroboscopic evolution is described by repeated ap-plication of the Floquet operator. As this is a unitary opera-tor,
Fuf±l = es−i2pf±/k–duf±l. s8d f±are the Floquet quasienergies. Thus at a time which is n times the period of modulation, the state initially localized on +p0evolves to
ucsndl < ses−i2pnf+/k–duf+l + es−i2pnf−/k–duf−ld/Î2. s9d
Ignoring an overall phase and defining the separation be-tween Floquet quasienergies as
Df = f−−f+, s10d
one obtains
ucsndl < suf+l + es−i2pnDf/k–duf−ld/Î2. s11d
At
n = k–/s2Dfd s12d
periods, the state will form the antisymmetric superposition of Floquet states and thus is localized on the other fixed point
FIG. 4. Momentum distribu-tions as a function of the interac-tion time with the modulated standing wave calculated using numerical solutions of the Gross-Pitaevskii equation for the same parameters as Fig. 3. Initially at-oms have mostly negative mo-mentumst = 0.25Td. After
approxi-mately five modulation periods most atoms populate a state with positive momentum, therefore having undergone dynamical tun-neling. Corresponding experimen-tal data from Ref. [1] is also in-cluded as insets.
at −p0. In other words the atoms have tunneled from one of the fixed points to the other. This is reminiscent of barrier tunneling between two wells, where a particle in one well, in a superposition of symmetric and antisymmetric energy eigenstates, oscillates between wells with a frequency given by the energy difference between the eigenstates.
Tunneling can also occur when the initial state has signifi-cant overlap with two nonsymmetric states. For example, if the initial state is localized on two Floquet states, one local-ized inside the classical chaotic region and one inside the region of regular motion, a distinct oscillation in the strobo-scopic evolution of the mean momentum may be visible. The frequency of this tunneling oscillation depends on the spac-ing of the correspondspac-ing quasi-eigenenergies in the Floquet spectrum. In many cases multiple tunneling frequencies oc-cur in the stroboscopic evolution of the mean momentum some of which are due to tunneling between nonsymmetric states.
Quantum dynamical tunneling may be defined in that a particle can access a region of phase space in a way that is forbidden by the classical dynamics. This implies that it
crosses a Kolmogorov-Arnold-Moser (KAM) surface
[12,13]. The clearest evidence of dynamical tunneling can be
obtained by choosing the scaled Planck’s constant k– suffi-ciently small so that the atomic wave function is much smaller than the region of regular motion (the size of the wave function is given by k–). Furthermore, it should be cen-tered inside the region of regular motion. However, even if the wave packet is larger than the region of regular motion and also populates the classical chaotic region of phase space one can still analyze quantum-classical correspondence and tunneling. One assumes a classical probability distribution of point particles with the same size as the quantum wave func-tion and compares the classical evolufunc-tion of this point par-ticle probability distribution with the quantum evolution of the wave packet. A distinct difference between the two evo-lutions results from the occurrence of tunneling assuming that the quantum evolution penetrates a KAM surface vis-ibly.
C. Comparison
Using the Gross-Pitaevskii equation one can exactly simulate the experiment because the momentum or position representation is used. Therefore the theoretical simulation can be directly compared with the experimental result. In contrast Floquet states do not have a straightforward experi-mental intuitive analog. In the Floquet states analysis one can compare the quasienergy splitting between the tunneling Floquet state with the experimentally measured tunneling pe-riod. The occurrence of multiple frequencies in the experi-mentally observed tunneling oscillations might also be ex-plained with the presence of more than two dominant Floquet states, the tunneling frequencies being the energy splitting between different participating Floquet states. Using the Gross-Pitaevskii approach one can simulate the experi-ment with high precision. The same number of populated wells as in the experiment can be used and the turn-on of the standing wave can be simulated using an appropriate turn-on